An unstable abstract elementary class of modules: A variation of Paolini-Shelah's example

We construct a class $\widehat{K}$ of torsion-free abelian groups such that $\widehat{\mathbf{K}}=(\widehat{K},{\le }_p)$ is an abstract elementary class with $\mathrm{LS}(\widehat{\mathbf{K}})={\mathrm{\aleph }}_0$ such that: $(\cdot )$ $\widehat{\mathbf{K}}$ is not stable; $(\cdot )$ $\widehat{\mathbf{K}}$ has the joint embedding property and no maximal models, but does not have the amalgamation property; $(\cdot )$ $\widehat{\mathbf{K}}$ is $(<{\mathrm{\aleph }}_0)$-tame. The class we construct is a variation of [PaSh, Section 4] which isolates the core mechanism of the Paolini-Shelah construction.